8 COMPLETE HUBERT ALGEBRAS 361 Since 0?i£2)* = e*e* = e2ei> it is clear that (b) and (c) are equivalent. If (^|^2) = 0 then by (15.7.5.2) and (15J.5.5) we have 0 = (<?2|<?2) = (e1e2\e1e2) = Ik^ll2* so tnat (a) implies (b). Conversely, if e±e2 = 0, then (*i I e2) = (el I e2> = Oi I ^1^2) = 0, and therefore (b) implies (a). (15.8.5) Every left ideal I ^ {0} in A contains a self-adjoint idempotent ^0. Let x be a nonzero element of I. Then z = x*x ^ 0 (15.7.5.7), and z is a self-adjoint element of I. Multiplying z by a suitable nonzero real scalar, we may assume that ||£/(z)|| = 1, where U is the regular representation of A (15.8.1). Hence, by (15.8.1) and (11.5.3), ||tf(z2)||=l, and therefore || (7(z2M)|| = 1 by induction on n. On the other hand, we have \\U(zk+l)\\ = \\U(z)U(zk)\\ £ \\U(z)\\ • ||<7(z*)|| = ||t/(z*)|| for all A:, hence the sequence (||C/(zk)||) is decreasing. Since it has infinitely many terms equal to 1, it follows that || U(zk)\\ = 1 for all integers k, and hence that ||z*|| ^ If || U\\ for all k. We shall show that the sequence (z2fc) is a Cauchy sequence in the Hilbert space A. Let n, p be two integers >0, and put m = n+p. Then (z2m | z2n) = (z2pz2n | z2") = (zp+2w | zp+2n) = || U(zp) * z2M||2 and (z2m | z2m) = || U(zp) - zp+2w||2 ^ (zp+2n | zp+2n) = (z2m | z2") so that, for all ra > «, I/ 1| l/||2 g (z2m | z2m) ^ (z2m I z2w) g (z2" I z2n). This shows that the sequence (||z2n||2) is decreasing and has a limit a> 0; moreover, we have ||z2m _ Z2n||2 = (z2m | ^ _ 2(z2m | ^ + (z2« | ^ ^ ||z2n||2 - a which proves that (z2k) is a Cauchy sequence, as asserted. Hence the sequence (z2fe) has a limit e. By continuity, e2 = Iim(z4k) = e, and e* = lim(z*)Zfc = e k-* oo fc-»-oo because z is self-adjoint, and finally ez2 = Iimz2fe+2 = e, so that eel. Finally, fc~>00 since ||zfc|| ^ I/ 1| U\\ for all fc ^ 1, we have ||e|| > 0, and the proof is complete. t A self-adjoint idempotent e ^ 0 is said to be reducible if there exist two orthogonal self-adjoint idempotents ei9 e2, each nonzero, and such that e^e^+62- In this case, by (15.8.4), we have ee1=e1e = el and ee2 = &2 e = e2 • If e ^ 0 is not reducible, it is said to be irreducible.ary elements of A is bounded.